Understanding Virtual Nodes: Oversquashing and Node Heterogeneity

TL;DR

This paper provides a theoretical analysis of virtual nodes in message passing neural networks, highlighting their role in mitigating oversquashing and proposing a variant that improves sensitivity to node importance based on graph topology.

Contribution

It offers a precise characterization of how virtual nodes improve mixing and oversquashing, and introduces a novel variant that enhances node sensitivity for graph tasks.

Findings

Virtual nodes improve mixing abilities depending on topology.

Classical virtual nodes assign uniform importance to nodes.

Proposed variant enhances sensitivity and performance on graph tasks.

Abstract

While message passing neural networks (MPNNs) have convincing success in a range of applications, they exhibit limitations such as the oversquashing problem and their inability to capture long-range interactions. Augmenting MPNNs with a virtual node (VN) removes the locality constraint of the layer aggregation and has been found to improve performance on a range of benchmarks. We provide a comprehensive theoretical analysis of the role of VNs and benefits thereof, through the lenses of oversquashing and sensitivity analysis. First, we characterize, precisely, how the improvement afforded by VNs on the mixing abilities of the network and hence in mitigating oversquashing, depends on the underlying topology. We then highlight that, unlike Graph-Transformers (GTs), classical instantiations of the VN are often constrained to assign uniform importance to different nodes. Consequently, we propose a variant of VN with the same computational complexity, which can have different sensitivity to nodes based on the graph structure. We show that this is an extremely effective and computationally efficient baseline for graph-level tasks.